Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
Is an elderly or chronically ill person required to pay additional fidyah if they delay it beyond the first year?
An elderly person or someone permanently unable to fast must pay fidyah by feeding one needy person for each missed day.
However, if they delay paying fidyah beyond the first year, no additional fidyah is required.
This differs from someone who delays making up missed Ramadan fasts (qada) without a valid excuse until the next Ramadan begins—such a person is required to pay an additional fidyah for the delay.
I bought an apartment, and I am saving money to pay the remaining balance of its price — is zakāh obligatory on this money?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Zakāh is obligatory on all saved/hoarded wealth. Once the total wealth reaches the niṣāb (minimum threshold liable for zakah— equivalent to the value of 85 grams of pure 24-karat gold) and a full ḥawl (lunar year) has elapsed over it, zakāh becomes obligatory at a rate of (2.5%) — even if the wealth is earmarked for basic future needs, and even if the wealth's owner is in debt. Debt does not prevent the obligation of zakāh, nor is it deducted when calculating the zakāh due, based on the general and unrestricted wording of all the Qur'anic verses commanding zakāh, which draw no distinction between someone in debt and anyone else.
It is stated in ‘Umdat as-Sālik, from the books of the Shāfi‘ī school: "If someone possesses exactly the niṣāb, and owes an equivalent amount in debt, zakāh remains obligatory on what is in his possession, and the debt does not prevent this obligation" — meaning, the obligation of zakāh on what is presently in his possession.
The ḥawl (annual cycle) recognized for zakāh, as well as for the other sharī‘ah obligations that Allah, the Exalted, has imposed upon Muslims, is the Hijrī (lunar) year. And Allah, the Exalted, knows best.
Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.